Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients

Hartono, Yusuf; Kraaikamp, Cor; Schweiger, Fritz

doi:10.5802/jtnb.371

Cited by 28 publications

(23 citation statements)

References 8 publications

(2 reference statements)

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“…Arithmetic and ergodic properties of T E associated to this new continued fraction expansion were studied by Y. Hartono, C. Kraaikamp and F. Schweiger in [7]. They showed that T E has no finite invariant measure equivalent to the Lebesgue measure, but has infinitely many σ -finite, infinite invariant measures.…”

Section: Introductionmentioning

confidence: 95%

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Arithmetic and metric properties of Oppenheim continued fraction expansions

Fan

Wang

2007

Journal of Number Theory

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Section: Introductionmentioning

confidence: 95%

“…In 2002, Y. Hartono, C. Kraaikamp and F. Schweiger [7] introduced a new continued fraction algorithm with nondecreasing partial quotients, named Engel continued fraction (ECF) expansion. Just as the name suggests, the ECF expansion is originated from the classical Engel series expansion.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic and metric properties of Oppenheim continued fraction expansions

Fan

Wang

2007

Journal of Number Theory

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“…In 2002, Hartono, Kraaikamp and Schweiger [2] introduced the Engel continued fraction Then it follows that the partial quotients satisfy b 1 …”

Section: Introductionmentioning

confidence: 99%

“…It was given in [2] that lim n→∞ P n Q n = x. For the GCF, we have the corresponding arithmetic property for the convergent of x: Proposition 1.1.…”

Section: Introductionmentioning

confidence: 99%

Metrical properties for a class of continued fractions with increasing digits

Zhong

2008

Journal of Number Theory

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In 2002, Hartono, Kraaikamp and Schweiger introduced the Engel continued fractions (ECF), whose partial quotients are increasing. Later, Schweiger generalized it into a class of continued fractions with increasing digits and a parameter , called generalized continued fractions (GCF). In this paper, we will give some arithmetic properties of such an expansion, and show that the GCF holds similar metric properties with ECF under the condition that −1 < 1. But when it comes to the condition that = −1, it does not.

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“…Oppenheim showed that for arbitrary quadratic irrational numbers x, infinitely many periodic proper expansions exist with a period of length 1. A conjecture was formulated by Oppenheim, which was disproved in [10].…”

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confidence: 95%

Continued fraction expansions with variable numerators

2014

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A new continued fraction expansion algorithm, the so-called a/b-expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's α-expansions, Schweiger's odd-and even-continued fraction expansions, and the Rosen fractions), these a/b-expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension.

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Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients

Cited by 28 publications

References 8 publications

Arithmetic and metric properties of Oppenheim continued fraction expansions

Arithmetic and metric properties of Oppenheim continued fraction expansions

Metrical properties for a class of continued fractions with increasing digits

Continued fraction expansions with variable numerators

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